854 CHAPTER 8 Applications of Trigonometry The Cycloid The cycloid is a special case of the trochoid — a curve traced out by a point at a given distance from the center of a circle as the circle rolls along a straight line. If the given point is on the circumference of the circle, then the path traced as the circle rolls along a straight line is a cycloid, which is defined parametrically as follows. x =at −a sin t, y =a −a cos t, for t in 1 −H, H2 EXAMPLE 4 Finding Alternative Parametric Equation Forms Give two parametric representations for the equation of the parabola. y = 1x - 222 + 1 SOLUTION The simplest choice is to let x = t, y = 1t - 222 + 1, for t in 1-∞, ∞2. Another choice, which leads to a simpler equation for y, is x = t + 2, y = t2 + 1, for t in 1-∞, ∞2. S Now Try Exercise 35. EXAMPLE 5 Graphing a Cycloid Graph the cycloid. x = t - sin t, y = 1 - cos t, for t in 30, 2p4 ALGEBRAIC SOLUTION There is no simple way to find a rectangular equation for the cycloid from its parametric equations. Instead, begin with a table using selected values for t in 30, 2p4. Approximate values have been rounded as necessary. Plotting the ordered pairs 1x, y2 from the table of values leads to the portion of the graph in Figure 77 from 0 to 2p. GRAPHING CALCULATOR SOLUTION It is easier to graph a cycloid with a graphing calculator in parametric mode than with traditional methods. See Figure 78. t 0 p 4 p 2 p 3p 2 2p x 0 0.08 0.6 p 5.7 2p y 00.31 21 0 y x P(x, y) 1 2p p 1 0 2 Figure 77 0 0 4 2p Figure 78 Using a larger interval for t would show that the cycloid repeats this pattern every 2p units. S Now Try Exercise 41. The cycloid has an interesting physical property. If a flexible cord or wire goes through points P and Q as in Figure 79, and a bead is allowed to slide due to the force of gravity without friction along this path from P to Q, the path that requires the shortest time takes the shape of the graph of an inverted cycloid. Q P Figure 79 NOTE Hypotrochoids and epitrochoids, curves related to trochoids, are traced out by a point that is a given distance from the center of a circle that rolls not on a straight line, but on the inside or outside, respectively, of another circle. The classic Spirograph toy can be used to draw these curves.
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